hw4_6243_2003

hw4_6243_2003 - ) , w ( ) = ( v ( )) . Make sure that your...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Problem Set 4 1 Problem 4.1 ¯ Find a function V : R 3 R + which has a unique minimum at x = 0, and is strictly monotonically decreasing along all non-equilibrium trajectories of system x ˙ 1 ( t ) = x 1 ( t ) + x 2 ( t ) 2 , x ˙ 2 ( t ) = x 2 ( t ) 3 + x 3 ( t ) 4 , x ˙ 3 ( t ) = x 3 ( t ) 5 . Problem 4.2 System takes arbitrary continuous input signals v : [0 , ± ) R and produces contin- uous outputs w : [0 , ± ) R in such a way that the series connection of and the LTI system with transfer function G 0 ( s ) = 1 / ( s + 1), described by equations x ˙ 0 ( t ) = x 0 ( t ) + w ( t ) , w ( · ) = ( v ( · )) , has a non-negative storage function with supply rate 0 v, ¯ w 0 . v w ) . x 0 , ¯ w ) = ( ¯ x 0 )(¯ ¯ (a) Find at least one nonlinear system which fits the description. (b) Derive constraints to be imposed on the values G ( ) of a transfer function G ( s ) = C ( sI A ) 1 B 1 Posted October 1, 2003. Due date October 8, 2003
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 with a Hurwitz matrix A , which guarantee that x ( t ) 0 as t ± for every solution of x ˙( t ) = Ax ( t ) + Bw ( t ) , v ( t ) = Cx ( t
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ) , w ( ) = ( v ( )) . Make sure that your conditions are satised at least for one non-zero transfer func-tion G = G ( s ). Problem 4.3 For the pendulum equation y ( t ) + y + sin ( y ) = , nd a single continuously dierentiable Lyapunov function V = V ( y, y ) that yields the maximal region of attraction of the equilibrium y = y = 0. (In other words, the level set x R 2 : V ( { x ) < 1 } schould be a union of disjoint open sets, one of which is the attractor of the zero equilibrium, and V ( y ( t ) , y ( t )) schould have negative derivative at all points of except the origin.)...
View Full Document

Page1 / 2

hw4_6243_2003 - ) , w ( ) = ( v ( )) . Make sure that your...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online